72 research outputs found
Periods implying almost all periods, trees with snowflakes, and zero entropy maps
Let be a compact tree, be a continuous map from to itself,
be the number of endpoints and be the number of edges of .
We show that if has no prime divisors less than and has a
cycle of period , then has cycles of all periods greater than
and topological entropy ; so if is the least prime
number greater than and has cycles of all periods from 1 to
, then has cycles of all periods (this verifies a conjecture
of Misiurewicz for tree maps). Together with the spectral decomposition theorem
for graph maps it implies that iff there exists such that has
a cycle of period for any . We also define {\it snowflakes} for tree
maps and show that iff every cycle of is a snowflake or iff the
period of every cycle of is of form where is an odd
integer with prime divisors less than
Bouncing trimer: a random self-propelled particle, chaos and periodical motions
A trimer is an object composed of three centimetrical stainless steel beads
equally distant and is predestined to show richer behaviours than the bouncing
ball or the bouncing dimer. The rigid trimer has been placed on a plate of a
electromagnetic shaker and has been vertically vibrated according to a
sinusoidal signal. The horizontal translational and rotational motions of the
trimer have been recorded for a range of frequencies between 25 and 100 Hz
while the amplitude of the forcing vibration was tuned for obtaining maximal
acceleration of the plate up to 10 times the gravity. Several modes have been
detected like e.g. rotational and pure translational motions. These modes are
found at determined accelerations of the plate and do not depend on the
frequency. By recording the time delays between two successive contacts when
the frequency and the amplitude are fixed, a mapping of the bouncing regime has
been constructed and compared to that of the dimer and the bouncing ball.
Period-2 and period-3 orbits have been experimentally observed. In these modes,
according to observations, the contact between the trimer and the plate is
persistent between two successive jumps. This persistence erases the memory of
the jump preceding the contact. A model is proposed and allows to explain the
values of the particular accelerations for which period-2 and period-3 modes
are observed. Finally, numerical simulations allow to reproduce the
experimental results. That allows to conclude that the friction between the
beads and the plate is the major dissipative process.Comment: 22 pages, 10 figure
Topological chaos: what may this mean ?
We confront existing definitions of chaos with the state of the art in
topological dynamics. The article does not propose any new definition of chaos
but, starting from several topological properties that can be reasonably called
chaotic, tries to sketch a theoretical view of chaos. Among the main ideas in
this article are the distinction between overall chaos and partial chaos, and
the fact that some dynamical properties may be considered more chaotic than
others
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